A Surprising Property of Multidimensional Hamiltonian Systems; Application to Semiclassical Quantization of Phase Space

TL;DR

This paper links symplectic topology's properties to semiclassical quantization, introducing 'quantum blobs' as a more general framework than traditional phase space cells, with implications for understanding quantum states.

Contribution

It demonstrates that EBK quantization aligns with symplectic topology principles, introducing quantum blobs as a novel, more versatile concept for phase space analysis.

Findings

EBK quantization is equivalent to symplectic area quantization.

Quantum blobs generalize phase space cells in quantum statistics.

Symplectic topology provides new insights into semiclassical states.

Abstract

Symplectic topology has become a thriving area of research in mathematics and physics since Gromov's discovery in 1985 of a surprising property of canonical transformations (and hence of hamiltonian flows),"the principle of the symplectic camel". We exploit this property to show that the usually EBK quantization is equivalent to the physical assumption that the only allowewed semiclassical states are those with symplectic area nh+h/2. We introduce the terminology "quantum blobs" for these states. Quantum blobs are more general and powerful tools than the usual phase space cells of quantum statistics.